- Combine the wine and shallots in a medium saucepan. Bring to a boil and cook for 15-20 minutes, until the mixture has reduced to about 1/2 cup. Whisk in the butter and sugar; remove from heat and season with salt and pepper to taste.
- Heat an indoor grill to high heat. Brush the grates with oil, then place the mushrooms on the grill gill-side down. Brush the tops of the mushrooms with the cabernet sauce. Cook for about 5 minutes, then flip the mushrooms over. Brush the gill-side with additional sauce and cook for 5 minutes more. Flip the mushrooms one more time and brush the tops with the rest of the sauce. Once the mushrooms are easily pierced with a fork (the time depends on the thickness of the mushrooms, but most will be done after 15 minutes on the grill), top them each with a slice of cheese and transfer the mushrooms to a plate.
- Brush the cut sides of the bread with oil and sprinkle with rosemary. Place the bread, cut-side down, onto the grill and cook until lightly toasted, about 3 minutes. Top the bottom halves of the bread with the mushrooms, tomatoes, and arugula. Cover with the top halves of the bread.
If you don't have an indoor grill, a grill pan or an outdoor grill will work instead, but cooking times will vary.